L(s) = 1 | − 5.27i·2-s − 3i·3-s − 19.8·4-s − 15.8·6-s + 7i·7-s + 62.3i·8-s − 9·9-s + 34.7·11-s + 59.4i·12-s + 37.2i·13-s + 36.9·14-s + 170.·16-s − 10.5i·17-s + 47.4i·18-s + 58.5·19-s + ⋯ |
L(s) = 1 | − 1.86i·2-s − 0.577i·3-s − 2.47·4-s − 1.07·6-s + 0.377i·7-s + 2.75i·8-s − 0.333·9-s + 0.952·11-s + 1.43i·12-s + 0.795i·13-s + 0.704·14-s + 2.66·16-s − 0.150i·17-s + 0.621i·18-s + 0.707·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.582913207\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.582913207\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 2 | \( 1 + 5.27iT - 8T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.80iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 250. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 536. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28202195913364201281787630876, −9.346483852382342899316393469595, −8.904564292798051878369372190906, −7.71848753367446687275909667916, −6.37837933494822236795383319434, −5.10837873584692825721558446935, −4.01677790646131474900435915691, −3.00031539617151425531059670644, −1.87751376456170625084113361430, −0.963608566570048290128864327638,
0.67574010740331151210242140536, 3.40454739710177093488604277805, 4.47446653380665535120407296739, 5.21943972589373640909981174968, 6.32968178172018705885140431910, 6.90421590248875190456124920058, 8.139708618350462244097370190198, 8.554256718881911862224259976479, 9.704184229610351348786383116664, 10.22386282367167276439168571078